Advertisements
Advertisements
प्रश्न
Integrate the function in (sin-1x)2.
Advertisements
उत्तर
Let `I = int (sin^-1 x)^2 dx`
Put `sin^-1 x = theta`
⇒ x = sinθ
⇒ dx = cosθ dθ
∴ `I = int theta^2 cos theta d theta`
`= theta^2 int (cos theta) d theta - int (d/ (d theta) (theta^2) * int cos theta d theta) d theta`
`= theta^2 (sin theta) - int 2 theta (sin theta) d theta`
`= theta^2 sin theta - 2 int theta sin theta d theta + C`
`= theta^2 sin theta - 2 [theta * (- cos theta) - int 1 * (- cos theta) d theta] + C`
`= theta^2 sin theta + 2 theta cos theta - 2 int cos theta d theta + C`
`= theta^2 sin theta + 2 theta sqrt (1 - sin^2 theta) - 2 sin theta + C`
`= x (sin^-1 x)^2 + 2sin^-1 x sqrt (1 - x^2) - 2x + C`
APPEARS IN
संबंधित प्रश्न
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in `x^2e^x`.
Integrate the function in (x2 + 1) log x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : e2x sin x cos x
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
∫ log x · (log x + 2) dx = ?
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int 1/sqrt(x^2 - 9) dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int logx dx = x(1+logx)+c`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3 e^(x^2)dx`
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Trigonometric in the LIATE rule?
Evaluate \[\int e^x\sin x\,dx.\]
Repeated parts may be needed for which pair of integrals?
